Problem-based learning competence is one of the most important competencies of mathematical competence. Many researchers believe that this capacity is a factor that directly affects the development of students' competencies and qualities. Among the components of problem-based learning competencies, the generalization of the problem is one of the key factors. Especially generalization of problems about exponential functions and logarithmic functions is still an open issue, which is rarely interested in research. Our research is carried out to identify and analyze factors of problem-based learning competence through generalizing the actual problems on the topic of exponential and logarithmic functions in Vietnam. The questionnaire survey method with 47 high school teachers in Ho Chi Minh City, Vietnam, shows that 10.64% of teachers think that the teaching method of problem-based learning competence is of little importance; 25.53% think it is important and 63.83% think it is very important. We conducted experiments with 89 students and processed analytical data using SPSS software. From the results of qualitative and quantitative analysis, students showed interest and fascination with the generalized teaching method of the actual problem. The test results showed that the experimental class had a much higher score than the control class. The research also provides a table comparing the differences between problem-solving and discovery teaching with teaching to develop problem-based learning competence; the process of teaching generalizations as well as examples illustrating practical problems on the topic of exponents and logarithmic functions. Research results also show that students are excited, proactively discover and learn from real-life problems more than before and do more practical issues. The findings from the research have important implications in fostering problem-based learning competence through generalizing practical problems on the topic of exponents and logarithms for students.

In Vietnam today, the general education program has shifted from teaching focusing on content and knowledge to teaching towards the development of students' competencies and qualities. Therefore, the ability to predict and solve problems is of special concern. However, the high school graduation exam format is multiple-choice. This results in the students' relatively weak ability to reason and solve problems. Besides, students are quite surprised when getting used to real problems. Our research is done to analyze the factors that affect students' erroneous reasoning as well as identify common errors in real problems. We surveyed 47 teachers and 163 students in Ho Chi Minh City by questionnaire. Research has shown that 65.96% of teachers and 50.92% of students think that it is very necessary to practice and develop the ability to predict and solve problems through teaching to predict and correct mistakes in real-world problems, but most teachers do not fully understand the ability to predict and solve problems in teaching practical problems. Students' manifestations of each type of mistake in teaching the ability to predict and solve problems are not high and uneven. The research results also show that there are two main groups of factors affecting the ability to predict and correct mistakes through practical problems: first, the problem is not true to reality; Second, mistakes in reasoning to solve real problems. Findings from the research have important implications for renewing teaching methods in Vietnam.